An Estimate of the Maximal Operators Associated with Generalized Lacunary Sets

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dc.creatorKaragulyan, Grigor
dc.creatorLacey, Michael T
dc.date2004-04-02
dc.date.accessioned2026-07-07T06:24:47Z
dc.date.available2026-07-07T06:24:47Z
dc.descriptionLet $Ω$ be any set of directions (unit vectors) on the plane. In this paper we study maximal operator of the one dimensional maximal function computed in the directions of $Ω$ We are interested in extensions of lacunary sets of directions, to collections we call $N$--lacunary, for integers $N$. We proceed by induction. Say that $Ω$ is 1--lacunary iff $Ω$ is an ordinary lacunary set of vectors. Every $N+1$--lacunary set can be obtained from some $N$--lacunary $Ω_N$ adding some points to $Ω_N$. Between each two neighbor points $a,b\inΩ_N$ we can add a 1--lacunary sequence (finite or infinite). We show that for all $N$ lacunary sets $Ω$, $$ \|M_Ωf(x)\|_2\lesssim{}N \|f\|_2. $$ Observe that every set $Ω$ of $N$ points is $(C\log N)$--lacunary. We then obtain a Theorem of N. Katz \cite{Katz2}. Both the current inequality, and Katz' result are consequence of a general result of Alfonseca, Soria, and Vargas \cites{ASV2}. We offer the current proof as a succinct, self--contained approach to this inequality.
dc.description10 pages
dc.identifierhttps://arxiv.org/abs/math/0404027
dc.identifierhttp://arxiv.org/abs/math/0404027
dc.identifierIzv. Nats. Akad. Nauk Armenii Mat. 39 (2004), no. 1, 73--82; translation in J. Contemp. Math. Anal. 39 (2004), no. 1, 50--59 (2005) ( MR2168200 )
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/96664
dc.subjectClassical Analysis and ODEs
dc.titleAn Estimate of the Maximal Operators Associated with Generalized Lacunary Sets
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